Subadditive Ergodic Theorem Research Articles

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets’ multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman’s subadditive ergodic theorem.

Grid entropy in last passage percolation — A superadditive critical exponent approach

Working in the setting of i.i.d. last-passage percolation on RD with no assumptions on the underlying edge-weight distribution, we arrive at the notion of grid entropy — a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a 2020 paper by Bates. We partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates.

A gapped generalization of Kingman’s subadditive ergodic theorem

We state and prove a generalization of Kingman’s ergodic theorem on a measure-preserving dynamical system (X,F,μ,T) where the μ-almost sure subadditivity condition fn+m ≤ fn + fm◦Tn is relaxed to a μ-almost sure, “gapped,” almost subadditivity condition of the form fn+σm+m≤fn+ρn+fm◦Tn+σn for some non-negative ρn ∈ L1(dμ) and σn∈N∪{0} that are suitably sublinear in n. This generalization has a first application to the existence of specific relative entropies for suitably decoupled measures on one-sided shifts.

A global method for deterministic and stochastic homogenisation in BV

In this paper we study the deterministic and stochastic homogenisation of free-discontinuity functionals under linear growth and coercivity conditions. The main novelty of our deterministic result is that we work under very general assumptions on the integrands which, in particular, are not required to be periodic in the space variable. Combining this result with the pointwise Subadditive Ergodic Theorem by Akcoglu and Krengel, we prove a stochastic homogenisation result, in the case of stationary random integrands. In particular, we characterise the limit integrands in terms of asymptotic cell formulas, as in the classical case of periodic homogenisation.

A non-singular version of the Oseledeč ergodic theorem

AbstractKingman’s subadditive ergodic theorem is traditionally proved in the setting of a measure-preserving invertible transformation T of a measure space $(X, \mu )$ . We use a theorem of Silva and Thieullen to extend the theorem to the setting of a not necessarily invertible transformation, which is non-singular under the assumption that $\mu $ and $\mu \circ T$ have the same null sets. Using this, we are able to produce versions of the Furstenberg–Kesten theorem and the Oseledeč ergodic theorem for products of random matrices without the assumption that the transformation is either invertible or measure-preserving.

A shape theorem for the orthant model

We study a particular model of a random medium, called the orthant model, in general dimensions d≥2. Each site x∈Zd independently has arrows pointing to its positive neighbours x+ei, i=1,…,d with probability p and, otherwise, to its negative neighbours x−ei, i=1,…,d (with probability 1−p). We prove a shape theorem for the set of sites reachable by following arrows, starting from the origin, when p is large. The argument uses subadditivity, as would be expected from the shape theorems arising in the study of first passage percolation. The main difficulty to overcome is that the primary objects of study are not stationary which is a key requirement of the subadditive ergodic theorem.

Consistent estimation of the number of regimes in Markov-switching autoregressive models

Markov-switching models have become a popular tool in areas ranging from finance to electrical engineering. Determining the number of hidden regimes in such models is a key problem in applications. This paper proposes a strongly consistent estimator of the number of regimes for Markov-switching autoregressive models. By using subadditive ergodic theorem, law of iterated logarithm for martingales, together with results from information theory, we derive sufficient conditions to avoid underestimation as well as overestimation. In particular, we propose a modified information criterion, regime-switching information criterion (RSIC) which generates a simple and consistent model selection procedure. Finally, we conduct a Monte Carlo study to evaluate the efficacy of our procedure in finite sample and also compare the performance of RSIC with popular information criteria including AIC, BIC and HQC.

Subadditive and multiplicative ergodic theorems

A result for subadditive ergodic cocycles is proved that provides more delicate information than Kingman’s subadditive ergodic theorem. As an application we deduce a multiplicative ergodic theorem generalizing an earlier result of Karlsson–Ledrappier, showing that the growth of a random product of semicontractions is always directed by some horofunction. We discuss applications of this result to ergodic cocycles of bounded linear operators, holomorphic maps and topical operators, as well as a random mean ergodic theorem.

Subadditive Ergodic Theorem for Double Sequences

A functional ergodic theorem is proved for subadditive families of measurable functions $$(h_{k,n}\mid (k,n)\in D)$$ , where $$D\subset {\mathbb {N}}^2$$ is an additive semigroup and subadditivity means that $$h_{k_1+k_2,n_1+n_2}\le h_{k_1,n_1}+h_{k_2,n_2}\circ f^{n_1}$$ for some measure-preserving transformation f.

Analysis and Design of Robust Max Consensus for Wireless Sensor Networks

A novel distributed algorithm for estimating the maximum of the node initial state values in a network, in the presence of additive communication noise is proposed. Conventionally, the maximum is estimated locally at each node by updating the node state value with the largest received measurements in every iteration. However, due to the additive channel noise, the estimate of the maximum at each node drifts at each iteration and this results in nodes diverging from the true max value. Max-plus algebra is used as a tool to study this ergodic process. The subadditive ergodic theorem is invoked to establish a constant growth rate for the state values due to noise, which is studied by analyzing the max-plus Lyapunov exponent of the product of noise matrices in a max-plus semiring. The growth rate of the state values is upper bounded by a constant which depends on the spectral radius of the network and the noise variance. Upper and lower bounds are derived for both fixed and random graphs. Finally, a two-run algorithm robust to additive noise in the network is proposed and its variance is analyzed using concentration inequalities. Simulation results supporting the theory are also presented.

Stochastic Homogenisation of Free-Discontinuity Problems

In this paper we study the stochastic homogenisation of free-discontinuity functionals. Assuming stationarity for the random volume and surface integrands, we prove the existence of a homogenised random free-discontinuity functional, which is deterministic in the ergodic case. Moreover, by establishing a connection between the deterministic convergence of the functionals at any fixed realisation and the pointwise Subadditive Ergodic Theorem by Akcoglou and Krengel, we characterise the limit volume and surface integrands in terms of asymptotic cell formulas.

Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

AbstractWe formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similar to the classical lattice growth models, the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free-branching birth rate.

A characterization of linearly repetitive cut and project sets**Research supported by EPSRC grants EP/L001462, EP/J00149X, EP/M023540. HK also gratefully acknowledges the support of the Osk. Huttunen foundation.

For the development of a mathematical theory which can be used to rigorously investigate physical properties of quasicrystals, it is necessary to understand regularity of patterns in special classes of aperiodic point sets in Euclidean space. In one dimension, prototypical mathematical models for quasicrystals are provided by Sturmian sequences and by point sets generated by substitution rules. Regularity properties of such sets are well understood, thanks mostly to well known results by Morse and Hedlund, and physicists have used this understanding to study one dimensional random Schrödinger operators and lattice gas models. A key fact which plays an important role in these problems is the existence of a subadditive ergodic theorem, which is guaranteed when the corresponding point set is linearly repetitive.In this paper we extend the one-dimensional model to cut and project sets, which generalize Sturmian sequences in higher dimensions, and which are frequently used in mathematical and physical literature as models for higher dimensional quasicrystals. By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with cubical windows. We also prove that these are precisely the collection of such sets which satisfy subadditive ergodic theorems. The results are explicit enough to allow us to apply them to known classical models, and to construct linearly repetitive cut and project sets in all pairs of dimensions and codimensions in which they exist.

TASEP hydrodynamics using microscopic characteristics

The convergence of the totally asymmetric simple exclusion process to the solution of the Burgers equation is a classical result. In his seminal 1981 paper, Herman Rost proved the convergence of the density fields and local equilibrium when the limiting solution of the equation is a rarefaction fan. An important tool of his proof is the subadditive ergodic theorem. We prove his results by showing how second class particles transport the rarefaction-fan solution, as characteristics do for the Burgers equation, avoiding subadditivity. In the way we show laws of large numbers for tagged particles, fluxes and second class particles, and simplify existing proofs in the shock cases. The presentation is self contained.

Asymptotic shapes for ergodic families of metrics on Nilpotent groups

Let \Gamma be a finitely generated virtually nilpotent group. We consider three closely related problems: (i) convergence to a deterministic asymptotic cone for an equivariant ergodic family of inner metrics on \Gamma , generalizing Pansu's theorem; (ii) the asymptotic shape theorem for first passage percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of \Gamma ; (iii) the sub-additive ergodic theorem over a general ergodic \Gamma -action. The limiting objects are given in terms of a Carnot–Carathéodory metric on the graded nilpotent group associated to the Mal'cev completion of \Gamma .

A Cut-Invariant Law of Large Numbers for Random Heaps

Heap monoids equipped with Bernoulli measures are a model of probabilistic asynchronous systems. We introduce in this framework the notion of asynchronous stopping time, which is analogous to the notion of stopping time for classical probabilistic processes. A Strong Bernoulli property is proved. A notion of cut-invariance is formulated for convergent ergodic means. Then a version of the Strong law of large numbers is proved for heap monoids with Bernoulli measures. Finally, we study a sub-additive version of the Law of large numbers in this framework based on Kingman sub-additive Ergodic Theorem.

Multifractal analysis of the irregular set for almost-additive sequences via large deviations

In this paper we introduce a notion of free energy and large deviations rate function for asymptotically additive sequences of potentials via an approximation method by families of continuous potentials. We provide estimates for the topological pressure of the set of points whose non-additive sequences are far from the limit described through Kingman’s sub-additive ergodic theorem and give some applications in the context of Lyapunov exponents for diffeomorphisms and cocycles, and the Shannon-McMillan-Breiman theorem for Gibbs measures.

Stochastic homogenization of a nonconvex Hamilton–Jacobi equation

We present a proof of qualitative stochastic homogenization for a nonconvex Hamilton–Jacobi equations. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter, which have deterministic limits by the subadditive ergodic theorem and maximality.

THE RANGE OF TREE-INDEXED RANDOM WALK

We provide asymptotics for the range $R_{n}$ of a random walk on the $d$-dimensional lattice indexed by a random tree with $n$ vertices. Using Kingman’s subadditive ergodic theorem, we prove under general assumptions that $n^{-1}R_{n}$ converges to a constant, and we give conditions ensuring that the limiting constant is strictly positive. On the other hand, in dimension $4$, and in the case of a symmetric random walk with exponential moments, we prove that $R_{n}$ grows like $n/\!\log n$. We apply our results to asymptotics for the range of a branching random walk when the initial size of the population tends to infinity.

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.